A11 TRACON Controller reported an aircraft going below the MVA while waiting to be issued a visual approach due to high workload.

Narrative

I was working North Radar. I was getting busy with Aircraft Ys returning from the north; two Aircraft Zs landing/requesting approaches at EDF; 6 or so VFR requesting flight following/ VFR practice approaches; giving ANC releases; and getting EDF GCA PAR approaches. Aircraft X departs ANC to ZZZ. I issue vectors for a VA (Visual Approach). I need to get him in position for his VA while at the same time providing separation from the fights returning. I get Aircraft X where I want him and clear him on a VA to ZZZ. I go to call ZZZ and realize that the line is OTS. At the same time someone in the TRACON states what I missed; that ZZZ is closed. I turn Aircraft X around and ask his intentions. He now wants to go back to ANC and wants to know why ZZZ is closed. I clear to ANC and advise that we will check. A short while later the Front Line Manager says ZZZ is now open. I ask Aircraft X if he'd like to go back to ZZZ; which he does. I re-clear him and issue a 050 heading and 3;000 ft. I then get distracted by all the other requests. I don't realize where Aircraft X is until he says we need a VA. I look up and see he's at 3;000 in the 7;000 MVA. I clear him on the VA and switch him. At some point during this we try to open NL; [another sector/position]; but I feel too busy to give a brief. The controller comes over and starts to help out as an NHO; [hand-off]; without a briefing. This does help but he doesn't notice Aircraft X either. I don't recall the LA; [MSAW Low Altitude]; going off until after Aircraft X was already switched and cleared.This was my fault. I should have cleared Aircraft X on a VA as soon as the Aircraft Ys weren't a factor. I thought the 50 heading was a good heading that would have kept him in the longer portion of the 2;000 MVA. I was wrong; a 60 or 70 would have been better at his location. I felt busier that I'm sure I actually was. Opening NL would have helped but when it was tried; I felt too busy. I could have used my tools better. Made him blue; put a J-ring around him; put a leader line on; something to make him stand out and draw my attention. ANC and North Radar specifically have many little tasks that have to be completed to accomplish things. There are lots of tasks that pull focus away from the scope. During this session I had to look up a location on our IDS (Information Display System) chart to see if it was approved to go direct or if I needed to issue a heading out a gate. Those are the types of tasks that draw focus away from separation.